Estimating a survival function and predicting the lifetimes of future patients are important medical problems. Data in clinical studies is often censored because not all members of the study reach a prescribed endpoint before the conclusion of the study. In addition, more than a single time measurement is often of interest, for instance, the time until recurrence of a disease and the time until death may both be variables deemed important. Development of methods of statistical analysis for this multivariate censored data is the aim of this proposed research. The research will consist of the development, testing, and study of optimality properties for three new procedures, and the testing and study of the most promising existing technique of analysis. The procedures proposed are nonparametric since parametric forms can be quite restrictive in multiple dimensions. For each of these procedures the testing phase will include the analysis of actual biomedical censored data comparing the relative merits of the estimators. The superiority of multivariate analysis over the current univariate methods of analysis will also be indicated. The testing phase will also include a computer simulation study which will help to indicate the strengths and weaknesses of each of the estimators proposed, by allowing testing over a wide range of conditions where the correct answer is known. The first procedure proposed is a Bayesian estimate of the survival curve. The estimator is difficult to compute and no large sample optimality properties are known. Approximations to the estimate will be found which allow computation of the estimate for moderate sample sizes. This will be accomplished by splitting the data into three subtypes of censored data which will simplify the analysis. The large sample properties of the estimator will be studied utilizing a simplification of the multivariate probability distributions involved to univariate distributions. The second procedure is a Bayesian predictive method which generalizes a known univariate method. The method has the advantage of being more suitable for prediction of new observations than any of the currently used methods. Techniques of attack in the development of this estimator are somewhat speculative. The third method is a generalization of the univariate Kaplan-Meier estimator based on the redistribution of probability associated with censored observations. The techniques of nonparametric smoothing allow this redistribution to occur in a similar manner to the univariate case by utilizing information obtained from nearby data. Large sample properties of the estimator will be studied using methods applicable to the univariate estimator. Finally, large sample properties of the nonparametric maximum likelihood estimator will be studied. Preliminary study will be via simulation results designed to determine how the estimator behaves for reasonably large samples. The mathematical large sample properties will be developed in conjunction with similar studies on the Bayesian estimator since the estimators are similar.